Chiral Majorana fermion from quantum anomalous Hall plateau transition

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1 Chiral Majorana fermion from quantum anomalous Hall plateau transition Phys. Rev. B, 2015 王靖复旦大学物理系 Science,

2 Acknowledgements Stanford Biao Lian Quan Zhou Xiao-Liang Qi Shou-Cheng Zhang Tsinghua Yang Feng Xiao Feng Ke He Yayu Wang Qi-Kun Xue UCLA Xufeng Kou Qinglin He Kang L Wang 2

3 References [1] J. Wang, B. Lian, S. C. Zhang, Phys. Rev. B 89, (2014) [2] J. Wang, B. Lian, S. C. Zhang, Phys. Scr. T164, (2015) [3] J. Wang, Q. Zhou, B. Lian, S. C. Zhang, Phys. Rev. B 92, (2015) [4] X. Kou, L. Pan, J. Wang et al, Nat. Commun. 6, 8474 (2015) [5] Y. Feng, X. Feng, Y. Ou et al, Phys. Rev. Lett. 115, (2015) [7] B. Lian, J. Wang, S. C. Zhang, Phys. Rev. B 93, (2016) [8] J. Wang, Phys. Rev. B 94, (2016) [9] Q. L. He et al, Science 357, 294 (2017) [10] B. Lian, J. Wang, X. Sun, A. Vaezi, S. C. Zhang, arxiv: (2017) 3

4 Dirac equation and the anti-particle In 1928, Dirac unified Einstein s special theory of relativity with quantum mechanics, and introduced Dirac equation where are Dirac s anticommuting Gamma matrices. Dirac equation gives negative energy solutions, which led Dirac to predict the existence of anti-particle. In 1932, the positron, the anti-particle of the electron was discovery by CD Anderson in cosmic rays. 4

5 Majorana and his fermion In 1937, Ettore Majorana asked the question: can fermions be their own antiparticles? The Dirac equation is known to describe charged fermions: where are Dirac s anticommuting Gamma matrices. Majorana claimed if all are selected imaginary, on can make real, describing a charge neutral, spin ½ fermion, obeying majorana equation Ettore Majorana Gamma matrices in Majorana equation. 5

6 Properties of the Majorana fermion Neutrino could be a Majorana fermion, with Majorana mass term. Majorana fermion is essential for supersymmetry. Chiral Majorana fermion could exist in 1+1 and 9+1 dimensions, both essential for the superstring theory. With only fermion-number-parity conservation. Majorana fermion could arise as quasi-particles of topological states of quantum matter. Majorana fermion could be used for topological quantum computing. Search for hypothetical particles/wave Higgs boson, gravitational wave Majorana fermion Magnetic monopole Axion Dark matter particle 6

7 Particle = anti-particle A Majorana fermion is its own anti-particle 1 = 1 In other words: 2 = 0 In the sense of modular (clock) arithmetic like 11+14=25=1 in clock counting In particle physics, Majorana neutrino violates lepton number conservation by 2 In condensed matter physics, Majorana fermion are associated with superconductors, which also violates particle number conservation by 2. 7

8 Topological insulators and superconductors in 2D Full pairing gap in the bulk, gapless Majorana edge and surface states Chiral Majorana fermions = 1/2 Chiral fermions s-wave superconductor = 1/2 proximity = 1/2 magnetic doping massless Majorana fermions massless Dirac fermions Qi, Hughes, Raghu and Zhang, PRL,

9 Basic mechanism of the QAH effect Key point to get Quantum Anomalous Hall effect: spin polarized band inversion Qi, Wu & Zhang, PRB 74, (2006): general theory Liu et al, PRL, 101, (2008): HgMnTe Yu et al, Science 329, 61, 2010 : (BiCr) 2 Te 3 9

10 The model of the 2D topological insulator (BHZ, Science 2006) Square lattice with 4-orbitals per site: s, x y x y, s,, ( p ip,, ( p ip ), Nearest neighbor hopping integrals. Mixing matrix elements between the s and the p states must be odd in k. H eff ( k x, k y ) m Bk A( kx ik 2 ) h( k) 0 m( k) h( k) A(sin kx i sin k y A( k x h y ) ik m Bk 0 ( k) y 2 A(sin k ) x i sin k m( k) mb mb y ) d a a ( k) Similar to relativistic Dirac equation in 2+1 dimensions, with a mass term tunable by the sample thickness d! 0, 0, Edge state No edge state 10

11 QAH model: FM exchange field h H * eff ( k hk ( ) x, k y ) h( k) 0 h 0 ( k) 2 m Bk A kx iky ( ) A( k ik ) m Bk x y hexchange 2 m Bk A( k ) x iky ( k) 2 ( ) A kx iky m Bk ( k) m, m, m B the same sign m m B the opposite sign m B B 11

12 Gapped Dirac fermions on the surface, chiral fermions on the domain wall 2D system Ferromagnetic Topological Insulating FM chiral fermion 3D topological insulator FM QAH can be realized in ferromagnetic TI (Qi, Hughes, Zhang, PRB 2008) 12

13 Phase diagram: QAH in a magnetic TI thin film Key message: strong FM ordering Condition for QAH: M V M J. Wang, B. Lian, S. C. Zhang, Phys. Scr. T164, (2015) 13

14 Experimental observation of the QAHE in Cr-BiSbTe3 (Tsinghua 2013, RIKEN, UCLA, Stanford, MIT, Princeton and PSU ) 14

15 QAH plateau transition: Chalker-Cottington model and effectively tune magnetic exchange coupling Magnetic TI: Two copy of Dirac model with opposite Chern number Three kinds of disorder potential: Network of chiral edge states at random magnetic domain walls J. Wang et al, Phys. Rev. B 89, (2014) 15

16 Critical behavior in QAH plateau transitions Theoretical Prediction Two copy of Dirac model with opposite Chern number 1. At coercivity field, and at low enough temperature, plateaus at 0! 2D dirty limit J. Wang, B. Lian, SCZ, PRB 89, (2014) 16

17 Observation of zero Hall plateau state (Tsinghua, UCLA, 2015, RIKEN 2017) 17

18 Chiral topological superconductivity in 2D 1 chiral fermion=2 identical majorana fermion X L Qi et al, PRB 82, (2010) 18

19 Model for superconductor proximity coupled QAH effect in magnetic TI The topological properties of H + characterize a px+/-ipy superconductor, depending on mass term Jing Wang et al, PRB 92, (2015) 19

20 Simple criteria: chiral TSC emerges at the QAH plateau transition Phase boundary: Competetion between,, Usually λ Δ, m 0 m 0 < λ QAH effect λ ± m 0 < Δ Chiral TSC 20

21 Phase diagram and realization of TSC in magnetic TI 1. Finite chemical potential. 2. Top and bottom surface better have coupling, otherwise fine tuning of chemical potential into gap is needed. 3. SC proximity only to one surface. (top and bottom have different SC pairing order). J. Wang et al, PRB 92, (2015) 21

22 Smoking gun of chiral edge Majorana fermion: transport signature 1. QAH and N=2 TSC interface No backscattering Edge current perfectly transmitted 2. QAH and N=1 TSC One chiral majorana complete backscattering The other chiral majorana perfectly transmitted S. B. Chung et al, PRB 83, (2011), J. Wang et al, PRB 92, (2015) 22

23 Smoking gun: transport signature half-integer conductance plateau at the coercive field ½ conductance plateau for N=1 TSC and chiral Majorana edge state J Wang et al, PRB 92, (2015) 23

24 Stability of Chiral TSC against disorder Quantum percolation theory in D class with particle hole symmetry Critical scaling: 0 1/2, A class 1/2 1, D class B. Lian, J. Wang, X. Sun, A. Vaezi, S. C. Zhang, arxiv: (2017) 24

25 Experimental signature of chiral Majorana edge state (UCLA & Stanford, 2017) Science 357, 294 (2017) 8 mm 0.6 mm 25

26 Experimental signature of chiral Majorana edge state (UCLA & Stanford, 2017) Science 357, 294 (2017) 26

27 σ 13 measurement instead of σ 12 THEORY EXPERIMENT 27

28 Recent comments online from academic community 28

29 Recent comments online from academic community Quantum phase transition of chiral Majorana fermion in the presence of disorder, B. Lian, J. Wang, X. Sun, A. Vaezi, S. C. Zhang, arxiv: (2017) 29

30 Summary 1. Chiral topological superconductor can be realized from superconductor proximity coupled quantum anomalous Hall state. 2. Tunable parameters, such as magnetism and hybridization gap make magnetic topological insulator a good platform for chiral TSC. 3. Experimental observation of the ½ plateau as a compelling evidence of chiral Majorana fermion. 4. Hybrid topological materials host a lot of interesting topological phenomena. Thank you for your attention! 30